Friedman number

A Friedman number is an integer which, in a given base, is the result of an expression using all its own digits in combination with any of the four basic arithmetic operators (+, −, ×, ÷) and sometimes exponentiation. For example, 347 is a Friedman number since 347 = 7^3 + 4.

The first few base 10 Friedman numbers are:
25, 121, 125, 126, 127, 128, 153, 216, 289, 343, 347, 625, 688, 736, 1022, 1024, 1206, 1255, 1260, 1285, 1296, 1395, 1435, 1503, 1530, 1792, 1827, 2048, 2187, 2349, 2500, 2501, 2502, 2503, 2504, 2505, 2506, 2507, 2508, 2509, 2592, 2737, 2916, 3125, 3159 (sequence http://oeis.org/A036057 )

A nice Friedman number is a Friedman number where the digits in the expression can be arranged to be in the same order as in the number itself. For example, we can arrange 127 = 2^7 − 1 as 127 = −1 + 2^7.

The first nice Friedman numbers are:
127, 343, 736, 1285, 2187, 2502, 2592, 2737, 3125, 3685, 3864, 3972, 4096, 6455, 11264, 11664, 12850, 13825, 14641, 15552, 15585, 15612, 15613, 15617, 15618, 15621, 15622, 15623, 15624, 15626, 15632, 15633, 15642, 15645, 15655, 15656, 15662, 15667, 15688, 16377, 16384, 16447, 16875, 17536, 18432, 19453, 19683, 19739 http://oeis.org/A080035

http://en.wikipedia.org/wiki/Friedman_number


Determine why each of these is a Friedman number. You might also have fun confirming that 123456789 and 987654321 are Friedman numbers. Find some more Friedman numbers. What else can you show about Friedman numbers? If F(n) is the number of Friedman numbers less than n, can you show F(n)/n --> 1? or even disprove F(n)/n --> 0?


http://www2.stetson.edu/~efriedma/mathmagic/0800.html

100255 – Friedman number
103823 – nice Friedman number